A040027 -id:A040027 - cp's OEIS frontend

A345077 a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * a(k-1).

1, 6, 48, 414, 3876, 38946, 416808, 4722774, 56379756, 706236426, 9250945008, 126342991614, 1794459834036, 26445918969906, 403610795535288, 6367606516836774, 103683034842399996, 1739933892930544986, 30052751213767045248, 533635421576480845134, 9730601644306627161156

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Cf. A040027, A094419, A144223, A343523, A343975, A344735, A344840, A345078, A345081.

a[0] = 1; a[n_] := a[n] = 6 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1 + 6 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + 6 * x * A(x/(1 - x)) / (1 - x)^2.

A345078 a(0) = 1; a(n) = 7 * Sum_{k=1..n} binomial(n,k) * a(k-1).

Original entry on oeis.org

1, 7, 63, 609, 6349, 70693, 835051, 10408335, 136290371, 1867933865, 26712000161, 397487932457, 6140285212915, 98264596199651, 1626101133819855, 27779382241071769, 489188555650420493, 8867962363328434205, 165284825277198034611, 3163858565498874214559, 62133992974174011252635

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Cf. A040027, A144263, A238464, A343523, A343975, A344735, A344840, A345077, A345081.

a[0] = 1; a[n_] := a[n] = 7 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1 + 7 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + 7 * x * A(x/(1 - x)) / (1 - x)^2.

A345081 a(0) = 1; a(n) = 8 * Sum_{k=1..n} binomial(n,k) * a(k-1).

Original entry on oeis.org

1, 8, 80, 856, 9824, 119912, 1547376, 21007992, 298874496, 4440618120, 68706037904, 1104224971416, 18394192882336, 316974497161384, 5640790811468976, 103503851543959224, 1955546066369814208, 37994858794236710088, 758272809049577019600, 15527828509092566876888

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Cf. A040027, A221159, A238465, A343523, A343975, A344735, A344840, A345077, A345078.

a[0] = 1; a[n_] := a[n] = 8 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1 + 8 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + 8 * x * A(x/(1 - x)) / (1 - x)^2.

A352682 Array read by ascending antidiagonals. A(n, k) = (n-1)*Gould(k-1) + Bell(k) for n >= 0 and k >= 1, A(n, 0) = 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 3, 5, 6, 1, 4, 4, 8, 15, 21, 1, 5, 5, 11, 24, 52, 82, 1, 6, 6, 14, 33, 83, 203, 354, 1, 7, 7, 17, 42, 114, 324, 877, 1671, 1, 8, 8, 20, 51, 145, 445, 1400, 4140, 8536, 1, 9, 9, 23, 60, 176, 566, 1923, 6609, 21147, 46814

Offset: 0

Peter Luschny, Mar 28 2022

The array defines a family of Bell-like sequences. The case n = 1 are the Bell numbers A000110, case n = 0 is A032347 and case n = 2 is A038561. The n-th sequence r(k) = T(n, k) is defined for k >= 0 by the recurrence r(k) = Sum_{j=0..k-1} binomial(k-1, j)*r(j) with r(0) = 1 and r(1) = n.

			Array starts:
n\k 0, 1,  2,  3,  4,   5,    6,    7,     8,      9, ...
---------------------------------------------------------
[0] 1, 0,  1,  2,  6,  21,   82,  354,  1671,   8536, ... A032347
[1] 1, 1,  2,  5, 15,  52,  203,  877,  4140,  21147, ... A000110
[2] 1, 2,  3,  8, 24,  83,  324, 1400,  6609,  33758, ... A038561
[3] 1, 3,  4, 11, 33, 114,  445, 1923,  9078,  46369, ... A038559
[4] 1, 4,  5, 14, 42, 145,  566, 2446, 11547,  58980, ... A352683
[5] 1, 5,  6, 17, 51, 176,  687, 2969, 14016,  71591, ...
[6] 1, 6,  7, 20, 60, 207,  808, 3492, 16485,  84202, ...
[7] 1, 7,  8, 23, 69, 238,  929, 4015, 18954,  96813, ...
[8] 1, 8,  9, 26, 78, 269, 1050, 4538, 21423, 109424, ...
[9] 1, 9, 10, 29, 87, 300, 1171, 5061, 23892, 122035, ...

Rows: A032347, A000110 (Bell), A038561, A038559, A352683.
Diagonals: A352684 (main).
Cf. A040027 (Gould), A352686 (subtriangle).
Compare A352680 for a similar array based on the Catalan numbers.

function BellRow(m, len)
    a = m; P = BigInt[1]; T = BigInt[1]
    for n in 1:len
        T = vcat(T, a)
        P = cumsum(vcat(a, P))
        a = P[end]
    end
T end
for n in 0:9 BellRow(n, 9) |> println end

alias(PS = ListTools:-PartialSums):
BellRow := proc(n, len) local a, k, P, T;
a := n; P := [1]; T := [1];
for k from 1 to len-1 do
   T := [op(T), a]; P := PS([a, op(P)]); a := P[-1] od;
T end: seq(lprint(BellRow(n, 10)), n = 0..9);

nmax = 10;
BellRow[n_, len_] := Module[{a, k, P, T}, a = n; P = {1}; T = {1};
   For[k = 1, k <= len - 1, k++,
      T = Append[T, a]; P = Accumulate[Join[{a}, P]]; a = P[[-1]]];
   T];
rows = Table[BellRow[n, nmax + 1], {n, 0, nmax}];
A[n_, k_] := rows[[n + 1, k + 1]];
Table[A[n - k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 15 2024, after Peter Luschny *)

Given a list T let PS(T) denote the list of partial sums of T. Given two list S and T let [S, T] denote the concatenation of the lists. Further let P[end] denote the last element of the list P. Row n of the array with length k can be computed by the following procedure:
     A = [n], P = [1], R = [1];
     Repeat k-1 times: R = [R, A], P = PS([A, P]), A = [P[end]];
     Return R.

A038561 Left-hand border of triangle A046937.

Original entry on oeis.org

1, 2, 3, 8, 24, 83, 324, 1400, 6609, 33758, 185136, 1083233, 6726366, 44130128, 304741623, 2207682188, 16729947276, 132281116715, 1088831511000, 9311082630620, 82569723552561, 758057178490082, 7194283782101844, 70481938088367569

Offset: 0

Henry Gould, N. J. A. Sloane

For n>1: a(n) is the number of entries in the last blocks of all set partitions of [n]. a(3) = 8 because the number of entries in the last blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 3+1+1+2+1 = 8. - Alois P. Heinz, May 08 2017

H. W. Gould, A linear binomial recurrence and the Bell numbers and polynomials, preprint, 1998

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968

A040027(n) + B(n), where B(n) = Bell numbers A000110.
Related to A000110, A040027, A038559, A038560.
Column k=1 of A286416 (for n>1).

a038561 = head . a046937_row  -- Reinhard Zumkeller, Jan 06 2014

A038561List := proc(m) local A, P, n; A := [1,2]; P := [1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([A[-1], op(P)]);
A := [op(A), P[-1]] od; A end: A038561List(24); # Peter Luschny, Mar 24 2022

a[0, 0] = 1; a[1, 0] = 2; a[n_, 0] := a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; a[n_] := a[n, 0]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 06 2013 *)

G.f. A(x) satisfies: A(x) = 1 + x * (1 + A(x/(1 - x)) / (1 - x)). - Ilya Gutkovskiy, Jun 30 2020

A123346 Mirror image of the Bell triangle A011971, which is also called the Pierce triangle or Aitken's array.

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 15, 10, 7, 5, 52, 37, 27, 20, 15, 203, 151, 114, 87, 67, 52, 877, 674, 523, 409, 322, 255, 203, 4140, 3263, 2589, 2066, 1657, 1335, 1080, 877, 21147, 17007, 13744, 11155, 9089, 7432, 6097, 5017, 4140, 115975, 94828, 77821, 64077, 52922, 43833, 36401, 30304, 25287, 21147

Offset: 0

N. J. A. Sloane, Oct 14 2006

a(n,k) is k-th difference of Bell numbers, with a(n,1) = A000110(n) for  n>0, a(n,k) = a(n,k-1) - a(n-1, k-1), k<=n, with diagonal (k=n) also equal to Bell numbers (n>=0). - Richard R. Forberg, Jul 13 2013
From Don Knuth, Jan 29 2018: (Start)
If the offset here is changed from 0 to 1, then we can say:
a(n,k) is the number of equivalence classes of [n] in which 1 not equiv to 2, ..., 1 not equiv to k.
In Volume 4A, page 418, I pointed out that a(n,k) is the number of set partitions in which k is the smallest of its block.
And in exercise 7.2.1.5--33, I pointed out that a(n,k) is the number of equivalence relations in which 1 not equiv to 2, 2 not equiv to 3, ..., k-1 not equiv to k. (End)

			Triangle begins:
    1
    2   1
    5   3   2
   15  10   7  5
   52  37  27 20 15
  203 151 114 87 67 52
  ...

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 418).

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
A. Dil, Veli Kurt, Investigating Geometric and Exponential Polynomials with Euler-Seidel Matrices, J. Int. Seq. 14 (2011) # 11.4.6
Don Knuth, Email to N. J. A. Sloane, Jan 29 2018
Eric Weisstein's World of Mathematics, Bell Triangle.

Cf. A011971. Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, A011968, A011969, A046934, A011972, A094577, A095149, A106436, A108041, A108042, A108043.

a123346 n k = a123346_tabl !! n !! k
a123346_row n = a123346_tabl !! n
a123346_tabl = map reverse a011971_tabl
-- Reinhard Zumkeller, Dec 09 2012

a[n_, k_] := Sum[Binomial[n - k, i - k] BellB[i], {i, k, n}];
Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 03 2018 *)

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A123346_list = blist = [1]
for _ in range(2*10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A123346_list += reversed(blist)
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

a(n,k) = Sum_{i=k..n} binomial(n-k,i-k)*Bell(i). - Vladeta Jovovic, Oct 14 2006

More terms from Alexander Adamchuk and Vladeta Jovovic, Oct 14 2006

A351756 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 2x)) / (1 - 2x)^2.

Original entry on oeis.org

1, 1, 5, 23, 119, 709, 4749, 35031, 281271, 2438565, 22673021, 224739303, 2363075191, 26246762213, 306830932749, 3763323446487, 48292462190743, 646763208308421, 9020009372203965, 130737162573013159, 1965798562640921879, 30613694640191725381

Offset: 0

Ilya Gutkovskiy, Feb 18 2022

nonn

Cf. A004211, A040027, A122704, A351757.

nmax = 21; A[] = 0; Do[A[x] = 1 + x A[x/(1 - 2 x)]/(1 - 2 x)^2 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k - 1] 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k-1) * 2^(k-1) * a(n-k).

A298804 Triangle T(n,k) (1 <= k <= n) read by rows: A046936 with rows reversed and offset changed to 1.

Original entry on oeis.org

0, 1, 1, 3, 2, 1, 9, 6, 4, 3, 31, 22, 16, 12, 9, 121, 90, 68, 52, 40, 31, 523, 402, 312, 244, 192, 152, 121

Offset: 1

N. J. A. Sloane, Jan 30 2018, following a suggestion from Don Knuth, Jan 29 2018

This is another version of Moser's version (A046936) of Aitken's array (A011971).
Although offset 0 is better for A011971 and A046936, for this version offset 1 is more appropriate.
Comments from Don Knuth, Jan 29 2018 (Start):
a(n,k) is the number of set partitions (i.e. equivalence classes) in which (i) 1 is not equivalent to 2, ..., nor k; and (ii) the last part, when parts are ordered by their smallest element, has size 1; (iii) that last part isn't simply "1". (Equivalently, n>1.)
It's not difficult to prove this characterization of a(k,n). For example, if we know that there are 22 partitions of {1,2,3,4,5} with 1 inequivalent to 2, and 6 partitions of {1,2,3,4} with
1 inequivalent to 2, then there are 6 partitions of {1,2,3,4,5} with 1 inequivalent to 2 and 1 equivalent to 3. Hence there are 16 with 1 equivalent to neither 2 nor 3.
The same property, but leaving out conditions (ii) and (iii), characterizes Pierce's triangular array A123346.  (End)

			Triangle begins:
0,
1, 1,
3, 2, 1,
9, 6, 4, 3,
31, 22, 16, 12, 9,
121, 90, 68, 52, 40, 31
523, 402, 312, 244, 192, 152, 121
...

Don Knuth, Email to N. J. A. Sloane, Jan 29 2018

Cf. A011971, A040027, A046936, A123346.

A351757 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 3x)) / (1 - 3x)^2.

Original entry on oeis.org

1, 1, 7, 43, 289, 2239, 19699, 192025, 2042971, 23520715, 291099349, 3849621019, 54110928355, 804827487493, 12619011606775, 207885167529523, 3587864566792753, 64705561315720135, 1216574535057705979, 23797327657083197113, 483390249416359706995

Offset: 0

Ilya Gutkovskiy, Feb 18 2022

nonn

Cf. A004212, A040027, A255927, A351756.

nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x/(1 - 3 x)]/(1 - 3 x)^2 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k - 1] 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k-1) * 3^(k-1) * a(n-k).

A342196 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^2 * a(k-1).

Original entry on oeis.org

1, 1, 5, 23, 155, 1355, 14371, 183911, 2781283, 48726355, 976903875, 22183097191, 565060532965, 16016170519017, 501714014484813, 17265124180702953, 649178961366102597, 26544344366333824055, 1175291769917975444817, 56133021061270139242637, 2881893164859601701738005

Offset: 0

Ilya Gutkovskiy, Mar 04 2021

nonn

Cf. A023998, A040027, A101514, A102221, A342197, A342198, A342199.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 20}]

cp's OEIS Frontend

A345077 a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * a(k-1).

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A345078 a(0) = 1; a(n) = 7 * Sum_{k=1..n} binomial(n,k) * a(k-1).

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A345081 a(0) = 1; a(n) = 8 * Sum_{k=1..n} binomial(n,k) * a(k-1).

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A352682 Array read by ascending antidiagonals. A(n, k) = (n-1)*Gould(k-1) + Bell(k) for n >= 0 and k >= 1, A(n, 0) = 1.

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A038561 Left-hand border of triangle A046937.

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A123346 Mirror image of the Bell triangle A011971, which is also called the Pierce triangle or Aitken's array.

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A351756 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 2*x)) / (1 - 2*x)^2.

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A298804 Triangle T(n,k) (1 <= k <= n) read by rows: A046936 with rows reversed and offset changed to 1.

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A351756 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 2x)) / (1 - 2x)^2.

A351757 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 3x)) / (1 - 3x)^2.